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# Mathematical properties of the Platonic solids

One can specify the following mathematical characteristics in each of the five Platonic solids:

1. The radius of the sphere circumscribing the polyhedron;

2. The radius of the sphere inscribed in the polyhedron;

3. The surface area of the polyhedron;

4. The volume of the polyhedron.

## Tetrahedron:  All four faces are equilateral triangles The radius of a circumscribed sphere of tetrahedron is ,where "a" is side length The radius of an inscribed sphere of tetrahedron is   The surface area of tetrahedron It is possible to represent the surface area of tetrahedron as an area of the model for better illustration. The volume of tetrahedron is  The height of the tetrahedron is determined by the following formula: The distance to the center of the base of the tetrahedron is determined by the formula: ## Octahedron: All eight faces are equilateral triangles The radius of a circumscribed sphere
of an octahedron is ,where "a" is side length The radius of an inscribed sphere of octahedron is   The surface area of octahedron is It is possible to represent the surface area of tetrahedron as an area of the model for better illustration. The volume of octahedron is ## Hexahedron (cube): All six faces are squares The radius of a circumscribed sphere
of cube ,where "a" is side length The radius of an inscribed sphere
of cube is   The surface area of cube is It is possible to represent the surface area of tetrahedron as an area of the model for better illustration. The volume of cube is ## Dodecahedron: All 12 faces are regular pentagon The radius of a circumscribed sphere
of dodecahedron ,where "a" is side length The radius of an inscribed sphere
of dodecahedron is   The surface area of dodecahedron is It is possible to represent the surface area of tetrahedron as an area of the model for better illustration. The volume of dodecahedron is ## Icosahedron: All 20 faces are equilateral triangles The radius of a circumscribed sphere of icosahedron ,where "a" is side length The radius of an inscribed sphere of icosahedron is   The surface area of icosahedron is It is possible to represent the surface area of tetrahedron as an area of the model for better illustration. the volume of icosahedron is ### Popular

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